General Theory of Relativity or the theory of relativistic gravitation is the one which describes black holes, gravitational waves and expanding Universe. The goal of the course is to introduce you into this theory. The introduction is based on the consideration of many practical generic examples in various scopes of the General Relativity. After the completion of the course you will be able to solve basic standard problems of this theory. We assume that you are familiar with the Special Theory of Relativity and Classical Electrodynamics. However, as an aid we have recorded several complementary materials which are supposed to help you understand some of the aspects of the Special Theory of Relativity and Classical Electrodynamics and some of the calculational tools that are used in our course. Also as a complementary material we provide the written form of the lectures at the website: https://math.hse.ru/generalrelativity2015
Do you have technical problems? Write to us: coursera@hse.ru

JQ

One of the most important courses ever taken and yet. still difficult to understand really well.

AB

Jan 29, 2017

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Demanding course with good overview of general relativity content.

À partir de la leçon

Cosmological solutions with non-zero cosmological constant

In this module we derive constant curvature de Sitter and anti de Sitter solutions of the Einstein equations with non-zero cosmological constant. We describe the geometric and causal properties of such space-times and provide their Penrose-Carter diagrams. We provide coordinate systems which cover various patches of these space-times.

Enseigné par

Emil Akhmedov

Associate Professor

Transcription

[SOUND] >> [MUSIC] >> Now we're going to continue with Poincare patch, and Poincare coordinates in the de Sitter space, and other possible solution of this equation that ABXBXB equals 2H minus to the second power is falling. We can write HX, well, this can be related to this. +HXD squared equal to 1-HXI+. The reason for the notations will become clear in a moment. So, we can define this to be equal to this, H tau+. And HX1 squared, + HXD-1 squared equal to HX plus i squared to h tow plus. So using this, we can write the following. Well, this can be now solved as HX0 = hyperbolic sine of H Tau+ + HX+I squared over 2 exponent of H of tau + HXi, this, and will be sold like this. HXi + e to the exponent of H tau+. Where i, both here, here, and here, and here is ranging from 1 to D-1, and finally, HXD, which solves this, together with this solve this equation, is minus hyperbolic assigned, of H tau + + HX + i squared divided by 2 exponent of H tau +. So, if we plug now these expressions into the ambient space Minkowskian metric, then we get metric induce on this space of the form, ds + squared = d tao + squared minus exponent of 2H tao + dx + vector squared. This guy is just this thing here. Now, and so the problem was this metric and with this coordinate. One can see immediately is that -X0 + XD as falls from here, and here is nothing but -1 over H exponent of HT+, which is always less than zero, which means that X0 is always greater than XD. So these coordinates do not cover entire. Unlike the previous coordinates, which we have been considering. They are called global coordinates and they cover entire de Sitter space, entire hyperboloid. These coordinates cover only half of it, so if those coordinates were covering all of this hyperboloid, these coordinates cover only upper half of it, we have to cut it by a plane. This and these coordinates cover only that half. And these coordinates are called polar coordinates. Well, polar coordinates of we will define in a moment. This is already almost coordinates. But this half of this de Sitter space is called Poincare patch. In fact, the other half can be covered by a similar coordinates with a bit different coronary change from this and then there we have the metric like this, DS- squared detail- squared, -2h tau-. DX- squared. So, these are expanding coordinates, expanding conquering patch. And this is contracting conqueror patch. Look, this factor says that special sections are expanding. So in this half, special sections are expanding, while in this half, special sections are contracting as the time goes by. So, on the ten rows, carve the diagram of this rectangle, rectangle, where. So, there are different choices for Poincare patches, but what we have, we did is that we have drawn this white like lines, which are cut by this plane which divides de Sitter space on two halves. So it's a like light lines, so this is a 90 degree angle here, and one can see that this is expanding for current patch. And this, remember, this is glued to this. And this is contracting Poincare patch. And in fact, we can cut out different expanding and contracting patches from the de Sitter space by different choices of this plane. Because this apex of this angle we can choose here. And this will be more or less the same. Situation equivalent to what we are discussing here. And one can see that constant tau plus slices are just this kind of lines. And constant time tau minus slices are these lines. And as time goes, this is tau- = -infinity. And this is tau- = +infinity. And this is tau+ = -infinity. And this is tau+ = +infinity. So we have expanding Poincare patch, partial sections are expanding, and here, it's contracting for a Poincare patch. Partial sections are contracting. It is convenient to introduce a so-called conformal time, here, so in this metric, in this metric, and in this metric, we make the following change of the coordinate. H eta plus minus = exponent of -+H tau +-. So for H eta +, we have here minus exponent of tau +. For eta- here, we have exponent of plus H times tau minus. In both cases, from this and this metric, after this change, we obtain the following metric, DS squared plus and minus look both the same, more or less, with one crucial difference, which I'll explain in a moment. So, D at the plus minus square minus DX plus minus square. So, this is Poincare coordinates for the Poincare patch. And constant eta, this is also constant eta slice, and this is also constant eta minus slice. But what is important, at past infinity eta+ is equal to infinity, at future infinity eta+ is equal to 0. This follows from here, because here, we have minus sign in the exponent. Tau equal to minus infinity gives us infinity here, and tau equals to plus infinity gives us zero, from here. And here, we have a reverse situation. That eta minus is equal to zero here, and ET minus equal to plus infinity here. So, basically expanding and contracting counter patch obviously can be obtained from each other by time reversal. Reversing the time from tal plus to minus tal minus, we obtain from expanding contracting patch and vice versa. And that is exactly what happens here. So, and the boundary between these two patches is just these two straight light rays. So, the past infinity of expanding upon Poincare patches, and future infinity of expanding upon Poincare patch is also light like. And one final two final points above the de Sitter space. Perhaps we can say them as follows one thing that in these coordinates the hyperbolic distance, and this other space in both cases, it looks the same. It was just a change, it was a use of ET plus or minus, and x plus and minus, so the hyperbolic distance between two points is like this. X2 squared divided by 2, eta 1 and eta 2. So, this is a hyperbolic distance, and basically, that's it. One probably final command is that one can cover entire the de Sitter space with this unique metric, with one eta parameter here, which coincide with eta minus here and eta plus here. But in convenience of this metric is that there is a jump of the values of eta as we cross this lines. So, these coordinates are not so convenient as the global coordinates to cover entire de Sitter space. One final point that I want to discuss about de Sitter space before continuing with anti de Sitter space is an notion of physical and moving volume. What is de Sitter space, basically? Well, consider a balloon, a balloon which tours plus infinity is expanding. So, from past infinity to the central de Sitter space to the neck of the de Sitter space is shrinking. So basically, at every given time we have a balloon of different ray years. And also to future infinity that is expanding, so this is three dimensional. This is basically if we are talking about two dimensional balloon. If we talk about circles, this is a two dimensional, de Sitter space. So, consider points spread over the de Sitter space. Say four points here. The same points. You see, as the time goes by, their numbers is not changed. Well, this is a dust. But their density is increasing while the source towards here is decreasing. But then starts, it increase again. And if we draw a line, a circle here, the same circle and trace its change in time, well, the area enclosed by this circle, volume enclosed by the, in case of multi dimensional balloons, this will be spheres. The volume enclosed by this will be physical volume. This is exactly physical volume. So, spatial physical volume. Remember that the center metric always in spatial sections has here. Either cosine squared HT over H squared D omega. Squared D-1. Or if in Poincare coordinates, it's either 1 over H eta+- squared DX +-squared. So, the volume measured with respect to this metric is the physical volume is exactly physical volume. So, we just DV, so the volume measure by this coordinates times the volume factor, the volume factor, which follows from here. So we have d, d-1, V falling from here, here. Multiplied by the appropriate power. D- one power. Of there, of this scale fracture. Well, for example, for this case, it's just consign HT divided by H. And for this case, it's 1 over H eta +-. So, this is physical volume, and it's changing according to how the physical distances are scaling. And the density, as we see, of the dust is changing accordingly like inverse of this. So, when it shrinks, the density is increasing. When it expands, the density is decreasing. But this volume, which is multiplied by, multiplying this factor, is called co-moving volume. Co-moving, volume so, somehow to obtain it, we subtract this factor. It's not. I don't understand how to draw this co-moving volume on this diagram. It's not the same as if you put it there, say, a solid object on the sphere, and it doesn't change its size. So, for example, we put something like this. It doesn't change its size, and it shrinks somehow and then expands back. No, this is not the co-moving volume because in that case, if this is a solid line, whose size is not changing in time, then the number of points will be changing within this line. For example, as it shrinks, the number of points coming into this line will be increasing. And it expands number of point is decreasing. The density per moving volume remains unchanged, number density. It is always four. Four points. You see the density with respect to this is always given by four points, but with respect to come moving, it doesn't change. With respect to the physical, four is attributed to the bigger distances, and that's the reason it is decreasing. I hope it's clear, what I mean. But this would finish our discussion of the de Sitter space. [SOUND]